TIME
COMPLEXITY
TURING
D.
OF
MULTIDIMENSIONAL
MACHINES
Yu.
Grigor'ev
UDC 510.52
It i s p r o v e d t h a t the w o r k of an i n d e t e r m i n a t e m - d i m e n s i o n a l
Turing machine with time com-
p l e x i t y t c a n b e s i m u l a t e d on an i n d e t e r m i n a t e k - d i m e n s i o n a l (k -< m) T u r i n g m a c h i n e with t i m e
c o m p l e x i t y t 1 - ( ~ / m ) + ( l / k ) + e ffor any e > 0).
Moreover,
the f o l l o w i n g g e n e r a l i z a t i o n to the m u l t i -
d i m e n s i o n a l c a s e of the f a m i l i a r t h e o r e m of H o p c r o f t , P a u l , a n d V a l i a n t is p r o v e d : the w o r k of
an m - d i m e n s i o n a l
Turing machine with time complexity t logi/mt
it(n) -> n] can be s i m u l a t e d on
an a d d r e s s m a c h i n e w o r k i n g w i t h t i m e c o m p l e x i t y t .
In the p r e s e n t p a p e r it is p r o v e d t h a t the w o r k of an i n d e t e r m i n a t e m - d i m e n s i o n a l T u r i n g m a c h i n e w i t h
t i m e c o m p l e x i t y t c a n be s i m u l a t e d on an i n d e t e r m i n a t e k - d i m e n s i o n a l (k -< m) T u r i n g m a c h i n e with t i m e c o m p l e x i t y t i + ( 1 / k ) ' f i / m ) + e (for a n y e > 0).
In a d d i t i o n , i t is r e m a r k e d t h a t the f a m i l i a r r e s u l t [1] on the t i m e g a i n in p a s s i n g f r o m T u r i n g m a c h i n e s
t o m a c h i n e s w i t h a r b i t r a r y a c c e s s to the m e m o r y
(in o t h e r w o r d s , r a n d o m a c c e s s m a c h i n e s , R A M , cf. [2]) c a n
b e g e n e r a l i z e d to t h e m u l t i d i m e n s i o n a l c a s e , m o r e p r e c i s e l y , to s i m u l a t e an m - d i m e n s i o n a l T u r i n g m a c h i n e
working with time complexity ttogl/mt
it(n) ~ n f o r any n], on a RAM w i t h t i m e c o m p l e x i t y t.
M o r e o v e r , the
l a s t s i m u l a t i o n can be e f f e c t e d on the a p p a r a t u s i n t r o d u c e d b y S l i s e n k o a n d c a l l e d in [3] an a d d r e s s m a c h i n e
(AM).
It i s a s p e c i f i c a t i o n of a RAM a n d i s c h a r a c t e r i z e d b y the f a c t t h a t in the c o u r s e of the e n t i r e w o r k to
i t s c o n c l u s i o n , the l e n g t h of t h e r e g i s t e r s u s e d d o e s n o t e x c e e d l o g 2 t + c , w h e r e t i s the t i m e of w o r k (the n u m b e r c is f i x e d f o r a g i v e n A M ) .
By D T M (ITM) we s h a l l d e n o t e a d e t e r m i n a t e
p r e c i s e d e f i n i t i o n , cf. [4]).
( i n d e t e r m i n a t e ) m u l t i d i m e n s i o n a l T u r i n g m a c h i n e (for the
In t h e c a s e when s o m e a s s e r t i o n i s t r u e b o t h f o r D T M and f o r I T M , we u s e the
n o t a t i o n T M , a n d h e r e it is u n d e r s t o o d t h a t e i t h e r a l l a p p a r a t u s e s c o n s i d e r e d in the g i v e n a s s e r t i o n a r e d e t e r minate or they are all indeterminate.
1.
In t h e f i r s t p o i n t of T h e o r e m 1, w h i c h i s p r o v e d b e l o w , t h e r e i s g i v e n an e s t i m a t e of the a m o u n t of
t i m e f o r s i m u l a t i n g ITM of h i g h e r d i m e n s i o n on a m a c h i n e of l o w e r d i m e n s i o n .
The m e t h o d u s e d is not s i m u -
l a t i o n o n - l i n e , in c o n t r a s t w i t h the m e t h o d a p p l i e d in [5], w i t h w h i c h t h e r e w a s o b t a i n e d an e s t i m a t e of the
a m o u n t of t i m e in l o w e r i n g the d i m e n s i o n on D T M .
We n o t e t h a t the e s t i m a t e o b t a i n e d in S e c . 1 f o r ITM i s
b e t t e r than the c o r r e s p o n d i n g e s t i m a t e f r o m [5] f o r D T M (which m e a n s a l s o the e s t i m a t e f o l l o w i n g f r o m [5] f o r
ITM).
The u p p e r b o u n d g i v e n in S e c . 1 i s s l i g h t l y w o r s e than the l o w e r b o u n d o b t a i n e d in [4] f o r o n - l i n e s i m u -
l a t i o n of TM on T M of l o w e r d i m e n s i o n .
N a m e l y (we u s e the n o t a t i o n of [5] a n d the c o r r e c t i o n of the r e s u l t of
[4] m a d e in [5]), it f o l l o w s f r o m [4] t h a t f o r any e > 0
~__L-
Translated from Zapiski Nauehnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta
i m . V. A. S t e k i o v a AN SSSR, V o l . 88, pp. 4 7 - 5 5 , 1979. O r i g i n a l a r t i c l e s u b m i t t e d M a r c h 23, 1976.
2290
0090-4104/82/2004-2290507o50
9 1982 P l e n u m P u b l i s h i n g C o r p o r a t i o n
The m e t h o d u s e d b e l o w a l l o w s one to do e v e n m o r e .
tion of the t r a j e c t o r i e s of the h e a d s .
In s t u d y i n g TM the q u e s t i o n a r i s e s of the c o n d e n s a -
T r a j e c t o r i e s can be " s p r e a d ~ o v e r a m u l t i d i m e n s i o n a l l a t t i c e .
~e
m e t h o d m a k e s it p o s s i b l e to s i m u l a t e the o r i g i n a l ITM in s u c h a way that the h e a d s s i m u l a t i n g ti~e ITM do not
l e a v e the l i m i t s of a cube with s m a l l edge.
We note that the m e t h o d of [5] a l s o a l l o w s one to get a s i m i l a r r e -
s u l t - to s i m u l a t e the w o r k of a TM with c a p a c i t y c o m p l e x i t y L a n d t i m e c o m p l e x i t y t , by a k - d i m e n s i o n a l TM,
w o r k i n g in a cube with edge L I/k-1, but the e s t i m a t e of t i m e h e r e is w o r s e t h a n - t L l / k - i
Upon s i m u l a t i n g
ITM on ITM of the s a m e d i m e n s i o n one c a n a c h i e v e c o n d e n s a t i o n c l o s e to o p t i m a l f o r p o w e r (with a r b i t r a r y
e x p o n e n t l a r g e r than one) l o s s of t i m e .
In p r o v i n g the s e c o n d point of T h e o r e m 1 u s i n g the s a m e m e t h o d it is shown that upon l o w e r i n g the d i m e n sion by o n e , one can get c o n d e n s a t i o n c l o s e to o p t i m a l , with a l m o s t no l o s s in t i m e .
T H E O R E M 1.
L e t k -< m be n a t u r a l n u m b e r s a n d e > 0.
T h e n f o r any m - d i m e n s i o n a l ITM M, w o r k i n g
with t i m e c o m p l e x i t y t and c a p a c i t y c o m p l e x i t y L, one can c o n s t r u c t
1.
a k - d i m e n s i o n a l ITM MI, w o r k i n g with t i m e c o m p l e x i t y ~ - ~ + e
in a cube with edge ~ §
2.
an (m + 1 ) - d i m e n s i o n a l ITM M2, w o r k i n g with t i m e c o m p l e x i t y ~ r ~
in a cube with e d g e ~
;
+$ ,
w h e r e M 1 a n d M 2 both have the s a m e output a s M.
( F o r the e a s e k = 1, point 1 of the t h e o r e m o v e r l a p s with the b a s i c r e s u l t of [6], e x t e n d e d to ITM.)
We give two a u x i l i a r y l e m m a s .
The f i r s t of t h e m is a m u l t i d i m e n s i o n a l g e n e r a l i z a t i o n of L e m m a 2 of [7]
and was u s e d in p r o v i n g a m u l t i d i m e n s i o n a l g e n e r a l i z a t i o n (whose f o r m u l a t i o n is given in [8]) of the b a s i c r e s u l t
of [71.
LEMMA 1.
L e t the h e a d s of the m - d i m e n s i o n a l ITM M on the p i e c e A of a zone (not n e c e s s a r i l y c o n -
n e c t e d ) , c o n t a i n i n g S > 2m + 1 c e l l s , o c c u r T t i m e s .
Then one can find a h y p e r p l a n e ~, o r t h o g o n a l to one of
the d i r e c t i o n s of the l a t t i c e , s u c h that
1) on e a c h of i t s s i d e s t h e r e a r e s i t u a t e d n o m o r e than
2(~)S
c e i l s of the p i e c e A;
2) the n u m b e r of p a s s a g e s of h e a d s of the ITM M (in h a n d l i n g the piece A) t h r o u g h (~ does not e x c e e d
c l T / S i / m , w h e r e c i d e p e n d s only on m a n d the n u m b e r of h e a d s of the ITM M.
Proof.
o t h e r left.
F o r e a c h of the m a x e s of the l a t t i c e by c o n v e n t i o n we c a l l one d i r e c t i o n on the a x i s r i g h t , the
We s i n g l e out the r i g h t (left) h y p e r p i a n e p a s s i n g t h r o u g h n o d e s of the l a t t i c e , o r t h o g o n a l to the
d i r e c t i o n c o n s i d e r e d and s u c h that on the left (right) side of it t h e r e a r e s i t u a t e d no m o r e than
~ ceils
of A.
The 2m h y p e r p i a n e s s i n g l e d out as a r e s u l t (for a l l m d i r e c t i o n s ) b o u n d a p a r a l l e i e p i p e d H, in which, by
v i r t u e of the choice of h y p e r p i a n e s , a r e s i t u a t e d not l e s s than ~--~-~y)~ c e i l s of A.
has length not l e s s t h a n k ~ 4 - ~ j
H e n c e one of the s i d e s of I1
. C o n s e q u e n t l y , one can find a h y p e r p l a n e ~, o r t h o g o n a [ to t h i s side and
i n t e r s e c t i n g 1], t h r o u g h which h e a d s of the ITM M p a s s not m o r e than e l t / S 1/m t i m e s .
LEMMA 2.
L e t P l = {1) . . . . .
a l and a 2 s u e h t h a t a j
Pi+l be o b t a i n e d f r o m Pi by r e p l a c i n g its m a x i m a l e l e m e n t a by s o m e two
>- ca (j = 1, 2, 1 / 2 >_ c > 0), w h e r e a 1 + a 2 = a .
Then any e l e m e n t of PN does not e x c e e d
1/eN.
By i n d u c t i o n on N one can p r o v e that if a 1 _> . . . _> a N a r e a l l e l e m e n t s of PN, then a N ~ cal.
Hence
2291
We p r o c e e d to the proof of T h e o r e m 1 (both points will be proved in para[iel).
We choose r sufficiently large~_ that one has ~ < d. = k - ~ . ( 2~m + 0
< ~-~+~r
I
< - k~ + s
in the case of point . and
< ~ in the case of point 2.
I
The simulation of the work of M wilt consist of the following,
We choose a (indeterm inate) hyperplane
with the p r o p e r t y indicated in L e m m a I, then we apply L e m m a 1 to the l a r g e r piece of the zone and thus r k
t i m e s (in the case of point 2 r m+l times) we apply L e m m a 1 (in both c a s e s if there r e m a i n s a piece of the zone
containing no m o r e than 2m + 1 c e i l s , then we no longer subdivide it).
Each time upon application of L e m m a 1
we subdivide indeterminately the l a r g e s t in n u m b e r of cells of the pieces of the zone.
Let us agree that the
l e t t e r c with indices will denote c o n s t a n t s , independent of t, L, s.
It wiU be proved by induction that the entire zone of the ITM M can be simulated in the m e m o r y of the
ITM M 1 ( o r M2), a c c o m m o d a t i n g it in a cube with side c3L (~ log~/kL (respectively, c3Lfl log~/m+lL), while to
each ceil of the active zone of the ITM M c o r r e s p o n d s its image, a cell of the m e m o r y of Ml (or M2), to which
t h e r e is attached a cube of side log~/kL (respectively, log~/m+lL) in which there is written the a d d r e s s of the
o r i g i n a l cell of the m e m o r y of the ITM M.
Let a piece of the active zone of the ITM M, consisting of s c e l l s , be divided in the way d e s c r i b e d above
in N = r k (respectively, N = r m+l) p i e c e s , containing s I -~ . . . -> s N, r e s p e c t i v e l y , active ceils.
L e m m a 2 to the collection of n u m b e r s ~sl/s . . . . .
We apply
SN/S) and we get that sl -< s / c N , here and later c = 1/(2m + 1).
By the inductive a s s u m p t i o n , the pieces of the zone of the ITM M, c o r r e s p o n d i n g to s i, are already packed in
cubes with sides c3s ~ log~/kL (respectively, c3s ~ log~/(m+l)L), so that the time r e q u i r e d by M I (or M 2) for
simulating the work of the ITM M on these pieces does not exceed c 2 t i s ~ - l / m log2 L (respectively, c2tiiog2s i 9
log 2 L); e 2 w i l l b e e h o s e n at
the end.
F o r pieces of the zone containing no more than 2m + I ceils, the inequali-
ties indicated f o r the lengths of the sides of the cubes can be satisfied at the expense of a suitable choice of c 3.
Since the pieces c o r r e s p o n d i n g to s i can be disconnected, one e s t i m a t e s the sum of the t i m e s n e c e s s a r y
for some head of the ITM M 1 (or M2), o v e r all intervals in which the head of the ITM M modeled by it are found
in a piece of the zone c o r r e s p o n d i n g to s i.
1Vioreover, one e s t i m a t e s that at the s t a r t of each such interval the e o r -
responding head of the ITM M I (or M2) is found in the image of the ceil in which at the s t a r t of this interval the
head of the ITM M modeled by it is situated.
The work of the ITM M 1 (or M 2) consists of steps of two types.
F i r s t l y , there is the direct simulation of
the work of the ITM M for steps at which the heads of the ITM M do not pass through the cuts made by the h y p e r planes (steps of the f i r s t type include consideration of the contents of ceils, the e n t r y of the new content, change
of state).
Secondly i s t h e s e a r c h for images of cells into which heads of the ITM M pass a f t e r intersecting cuts.
The latter will be effected indeterminately by s h o r t e s t paths, at the end of the s e a r c h it is only n e c e s s a r y to
verify that the a d d r e s s of the ceil [it is e n t e r e d in a cube with side iog~/kL (respectively, log 1/(m+l) ] is r e quired.
Cubes of the m e m o r y of the ITM M1 (or M 2) with sides c3s[~ log~/kL (respectively, c3s ~ log~/(m+l)L),
where 1 -< i -< N, can be packed into a cube with side c3rs ~ log~/kL (respectively, c3rsr log~/(m+l)L).
Lemma 2
--~/~
2292
0~0!/K.
S~
~--
ilk
Then by
and, respectively,
4
by v i r t u e of the c h o i c e of c~,/3, w h i c h p r o v e s the i n d u c t i v e s t e p on the l e n g t h of a s i d e of the cube of the m e m o r y
of the ITM M 1 (or M~).
It r e m a i n s to e s t i m a t e the t i m e .
In h a n d l i n g a p i e c e of the zone c o r r e s p o n d i n g to s , at a s t e p of the f i r s t
t y p e the ITM M 1 (or M S) b y the i n d u c t i v e assumption s p e n d s t i m e not g r e a t e r than
respectively,
At a step of the second
type the ITM
M s (or M S) speiids time not exceeding
respectively,
T, -- c,
!
w h e r e s l , tj (1 _< j _< N) a r e the n u m b e r of c e l t s a n d the t i m e s of h a n d l i n g t h e m on the I T M M in p i e c e s of the
zone w h i c h a r e c u t out by the h y p e r p l a n e s at the j - t h s t e p of the p r o c e s s d e s c r i b e d a b o v e .
The sum z~ C~IJ}/{ ~)
bounds (by Lemma 1) the number of steps in whose time cuts happen, and
e3s ~ l o g 2 L ( r e s p e c t i v e l y , e3s~ log2L) b o u n d s the n u m b e r of s t e p s of the ITlVI M~ (or M S) in the s e a r c h f o r the
i m a g e of the n e c e s s a r y c e l l a f t e r p a s s i n g t h r o u g h a cut.
S i n c e s / m s 9e N, one h a s
respectively,
4
&.
H e n c e f o r the ITM M s one h a s
T4+T/~C~JCS ~ov]~+r
z
({-5)
~o~,~<c~ts "~o~.~, the
last in-
e q u a l i t y is a c h i e v e d by a s u i t a b l e c h o i c e of e 2 [we note t h a t the c h o i c e of c3, c~ f o r the I T M M~ (or M 2) d i d not
d e p e n d on the c h o i c e of e~).
A n a l o g o u s l y f o r the ITM M S one h a s
a t the e x p e n s e of a s u i t a b l e c h o i c e of c 2.
The i n d u c t i v e s t e p on b o u n d i n g the t i m e is v e r i f i e d and T h e o r e m i i s
proved.
2.
In the s e c o n d s e c t i o n we g e n e r a l i z e to the m u l t i d i m e n s i o n a l c a s e one of the r e s u t t s of []1.
'The p r o o f
u s e s the m e t h o d p r o p o s e d in [1] and the m e t h o d of Sch6nhage [9] f o r s i m u l a t i n g in r e a l t i m e a TM by the
K o l m o g o r o v - U s p e n s k i i a l g o r i t h m [10]. In c o n n e c t i o n with the f a c t that the p r o o f h a s a e o m p i l a t i o n a l c h a r a c t e r ,
it is n o t r e c o u n t e d in g r e a t d e t a i l
T H E O R E M 2.
L e t the k - d i m e n s i o n a i TM M w o r k with t i m e c o m p l e x i t y not e x c e e d i n g t (t(n) ~> nlog~/kn).
Then t h e r e e x i s t s an A M R, w o r k i n g w i t h t i m e c o m p l e x i t y t / l o g / k t
and h a v i n g the s a m e output a s M.
2293
In the f i r s t s t a g e of the p r o o f , j u s t a s in [1], we t r a n s f o r m
that M' becomes block-respected,
(with l i n e a r d e l a y ) the TM M into a TM M' s o
cf. [1], with the b l o c k c i l o g i / k t ,
We d i v i d e the m e m o r y of M ' into c u b e s w i t h s i d e c 1 l o g t / k t .
w h e r e the c o n s t a n t c t w i l l be c h o s e n l a t e r .
The r e q u i r e m e n t of b e i n g b l o c k - r e s p e c t e d is t h a t
a l l t h e t i m e of w o r k of the TM M ' i s d i v i d e d into i n t e r v a l s of l e n g t h c l t o g l / k t , a n d in the c o u r s e of one i n t e r v a l
no h e a d i n t e r s e c t s b o u n d a r i e s of c u b e s .
To s a t i s f y the r e q u i r e m e n t of b l o c k - r e s p e c t we r e p l a c e e a c h h e a d of the TM M b y 3k h e a d s of the TM M'
a n d we a d d f u r t h e r f o r e a c h of t h e s e h e a d s a h e a d - i n d i c a t o r ,
w h i c h in s o m e c h o s e n cube with m a r k e d f a c e s w i l l
s i m u l a t e the p o s i t i o n of the h e a d in the c u b e a n d s i g n a l the t i m e when it g o e s p a s t the b o u n d a r y .
A l l the t i m e i n t e r v a l s of the w o r k of M ' a r e d i v i d e d into b a s i c a n d a u x i l i a r y .
In the t i m e of b a s i c i n t e r -
v a l s t h e r e o c c u r s s i m u l a t i o n of the w o r k of M, in t h e t i m e of a u x i l i a r y i n t e r v a l s h e a d s a s s u m e i n i t i a l p o s i t i o n s .
The i n i t i a l p o s i t i o n of the 3k h e a d s c o r r e s p o n d i n g to a h e a d of the T M M, at the b e g i n n i n g t i m e of a b a s i c i n t e r v a l is the f o l l o w I n g .
One h e a d , we s h a l l c a l l it c e n t r a l ,
is found in a c e l t of the TM M ' c o r r e s p o n d i n g to t h a t
c e l l of the TM M in w h i c h i t s h e a d b e i n g m o d e l e d i s found.
The r e m a i n I n g (3k - 1) h e a d s w h i c h we c a l l p e r i p h -
e r a l a r e found in n e i g h b o r i n g c u b e s on the b o u n d a r i e s in c e l l s c l o s e to t h a t c e l l in w h i c h the c e n t r a l h e a d is
found ( h e r e a n d l a t e r we d e s c r i b e the w o r k of the 3k h e a d s of the TM M ' , c o r r e s p o n d i n g to one h e a d of the TM
M, the w o r k of the r e m a i n i n g h e a d s i s s i m u l a t e d a n a l o g o u s l y ) .
In the t i m e of m o t i o n of the s i m u l a t e d h e a d of
the TM M i n s i d e a cube t h i s c o n d i t i o n is p r e s e r v e d .
S u p p o s e a t s o m e t i m e a h e a d of the TM M p a s s e s t h r o u g h the b o u n d a r y into one of the n e i g h b o r i n g c u b e s .
In t h i s c a s e the c o r r e s p o n d i n g c e n t r a l h e a d r e m a i n s on the b o u n d a r y , i t s r o l e s t a r t s to b e p l a y e d b y the c o r r e s p e n d i n g p e r i p h e r a l h e a d , a n d a l l the r e m a i n i n g (3k - 1) h e a d s a r e found at e a c h f o l l o w i n g m o m e n t in c l o s e s t
c e i l s to the n e w c e n t r a l h e a d .
Then t h e r e c a n a g a i n o c c u r a c h a n g e of c e n t r a l h e a d , e t c .
i n d i c a t e d a l l o w the b l o c k - r e s p e c t
The c o n s t r u c t i o n s
c o n d i t i o n to be s a t i s f i e d .
B y the c o n f i g u r a t i o n of the TM M' f o r the s t a r t of a t i m e i n t e r v a l we m e a n t h e c o n t e n t of a l l c u b e s in
w h i c h a t t h i s t i m e t h e r e i s found at l e a s t one h e a d of the TM M ' (these c u b e s w i l l be c a l l e d a c t i v e f o r t h i s i n t e r val), and neighborhood relations between active cubes.
By t h e c h o i c e of the c o n s t a n t c i one can a c h i e v e t h a t
t h e n u m b e r of a l l p o s s i b l e c o n f i g u r a t i o n s d o e s n o t e x c e e d c2t ~ f o r s o m e ~ < 1.
The p r e l i m i n a r y s t a g e of the w o r k of the A M R c o n s i s t s of the f o l l o w i n g .
A l l p o s s i b l e c o n f i g u r a t i o n s of
t h e TM M ' a r e e n t e r e d in the m e m o r y of AM (for e a c h c o n f i g u r a t i o n one n e e d s a f i x e d n u m b e r of r e g i s t e r s ) .
Next a f i x e d n u m b e r of r e g i s t e r s of the A M R a r e a v a i l a b l e f o r the e n t r y of the c o n t e n t of the a c t i v e c u b e s a f t e r
the w o r k of M ' in the c o u r s e of a t i m e i n t e r v a l of l e n g t h c, [ o g l / k t , the new s t a t e , the s i t u a t i o n of h e a d s , and
the i n d i c a t i o n of n e i g h b o r h o o d s of n e w a c t i v e c u b e s in r e l a t i o n to the o l d .
The p r o p e r s i m u l a t i o n of the w o r k of M ' u s e s w h a t w a s done b y the AM R in the p r e l i m i n a r y s t a g e a n d the
c o n s t r u c t i o n of S c h S n h a g e [9] (the a u t h o r c o n s i d e r e d it i n a p p r o p r i a t e to r e p r o d u c e in d e t a i l t h e c o n s t r u c t i o n of
[9]).
The m e m o r y of the TM M ' can b e d e s c r i b e d in t h e f o r m of a t r e e a n a l o g o u s to S c h S n h a g e ' s t r e e , but to
i t s l e a v e s a r e " a t t a c h e d " c u b e s w i t h s i d e c ~ l o g l / k t (the h a n d l i n g of t h i s t r e e i s e a s i l y s i m u l a t e d in r e a l t i m e on
an A M ) .
The c o n t e n t of the a c t i v e c u b e s is c h a n g e d in a c c o r d w i t h the p r e l i m i n a r y s t a g e , the SchSnhage t r e e
i s u s e d f o r f o r m i n g the c o n f i g u r a t i o n at the s t a r t of t h e n e x t t i m e i n t e r v a l - the t r e e with " a t t a c h e d " c u b e s
a l l o w s one to show t h e c o n t e n t a n d n e i g h b o r h o o d r e l a t i o n s b e t w e e n a c t i v e c u b e s .
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The time of work at the p r e l i m i n a r y stage is e s t i m a t e d as c3t a log~/kt, the time of p r o p e r simulation is
e s t i m a t e d as c t t / l o g t / k t .
Since the length of the entry of one active cube does not exceed ck[og~t, by lowering
c~ it is e a s y to satisfy the condition on the length of r e g i s t e r s f o r m u t a t e d in defining AM (of. [3]).
The author e x p r e s s e s thanks to A. O. S[isenko f o r i n t e r e s t in the work, S. V. Pakhamov f o r helpful disc u s s i o n s , and A. P. Bel'tyukov for valuable c o m m e n t s .
LITERATURE
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